Characterization Of Multiwavelet With Different Real Dilations And Translations

It is well known that an orthonormal wavelet (or, simply, a wavelet) is a function ψ(x) ∈ L2(R) such that the system {ψj,k(x) = 2j/2ψ(2jx- k)}, where j and k rang through the integers Z, forms an orthonormal basis of L2(R). In 1995, G. Gripenberg, E. Hernandez, X. Wang, G. Weiss, etc. gave the characterization of such an orthonormal wavelet in frequence. Moreover, C. K. Chui and X. L. Shi, in 2000, gave the characterization of orthonormal multiwavelets with an identical arbitrary real dilation factor a > 1 and an identical arbitrary real translation factor b > 0. Based on the result of C. K. Chui and X. L. Shi, in this thesis, the characterization of orthonormal multiwavelets with different real dilations and translations for the subspace L2E(R) of L2(R) is presented . The main results areTheorem A. Suppose that Ψ = ΨL = {Ψ1, ...,ΨL}, Ψ = ΨL - {Ψ1,...,ΨL} C L2E(R), al > 1, bl > 0, l = 1, ..., L and LΨ, LΨ ∈ L∞. Then (?) f, g ∈ DE, the series in (2.3) (P11 in the thesis) converge in the sense of (2.4) (Pll in the thesis), where, and f is compactly supported in E\{0} Theorem B. Under conditions of the theorem A, if P(f,g) = for f, g ∈DE,thenfor all a ∈ (?)L, where xE denotes tne cnaractenstic function of the set E.Theorem C. Let al > 1, bl > 0, ΨL = {Ψ1, ... ,ΨL} L2E(R) and Then ΨL is an orthonormal wavelet for LE(R) if and only ifFinally we end the thesis with some corollaries and examples.